FBX Exporter Animation Artefacts
I do not know what causes the problem, but I fixed it by restarting from scratch.  I appended just the mesh from the original blend file into a new one then I rebuilt the armatures.  The resulting exported fbx then did not suffer from the jitter problem.

Not a great answer, but I found the following which could be the reasons for not working in the first place:


Make sure scale is applied to everything before you start building the armatures.
It seems that the fbx exporter does not handle scaling in the animations well, so don't have any scaling in your animations.
The fbx exporter seems to ignore the objects' origins and uses the "world" origin instead.  To avoid any problems, set all the objects' origins to the "world" origin.


You may be able to get away with just applying these steps without having to rebuild the armature.

I have since then always followed these steps on new projects and it has worked every time.

I would like to hear from others if this works for them.